Acoustic Properties of Drums

Mode Diagrams for Circular Membranes (drumheads)

Adjected segment are moving in opposite directions

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The first 12 modes of an ideal circular membrane:

their nodal diagrams, mode designations, and their relative modal frequencies

The (0,1) Mode shows the fundamental mode shape of a vibrating circular membrane.

The mode number is designated as (0,1) since there are no nodal diameters, but one circlular node (the outside edge). Remember that a node is a point (or line) on a structure that does not move while the rest of the structure is vibrating. The (0,1) mode of a drum, such as a tympani, is excited when the drum head is struck at its center. When vibrating in this mode the membrane acts much like a monopole source, which radiates sound very effectively. Since it radiates sound so well when vibrating in this manner, the membrane quickly transfers its vibrational energy into radiated sound energy and the vibration dies away. The short duration (fraction of a second) of the (0,1) mode means that this mode does not contribute greatly to the musical tone quality of a drum. In fact, when struck at the center a tympani, or other large drum, produces a "thump" which decays quickly and with no definite pitch.

The (1,1) Mode with one nodal diameter and one circlular node (the outside edge).

The exact location of the nodal diameter depends on the homogeneity of the membrane and the initial conditions when the vibration starts. The frequency of the (1,1) mode is 1.593 times the frequency of the (0,1) mode. When vibrating in the (1,1) mode a circular membrane acts much like a dipole source; instead of pushing air away from the membrane like the (0,1) mode does, in the (1,1) mode one half of the membrane pushes air up while the other half sucks air down resulting in air being pushed back and forth from side to side. As a result, the (1,1) mode radiates sound less effectively than the (0,1) mode which means that it does not transfer its vibrational energy into radiated sound energy as quickly as the (0,1) mode and therfore, the (1,1) mode takes longer to decay. Because the (1,1) mode "rings" for a while, it contributes to the musical sound or pitch of a drum. When a tympani, or other large drum, is struck somewhere between the center and outer edge, the sound has a definite pitch which lingers for several seconds.

The (2,1) Mode: The third mode of a circular membrane is the (2,1) mode which has two nodal diameters (at right angles to each other) and one nodal circle (the outside edge).

The exact locations of the nodal diameters depend on the homogeneity of the membrane and the initial conditions when the vibration starts. The frequency of the (2,1) mode is 2.135 times the frequency of the (0,1) mode. When vibrating in the (2,1) mode a circular membrane acts much like a quadrupole source which is worse at radiating sound than the (1,1) dipole mode and much less effective at radiating sound than the (0,1) monopole mode. This means that the (2,1) transfers its vibrational energy into radiated sound energy much more slowly than the (1,1) and (0,1) modes and therefore takes longer to decay, and contributes to the musical pitch of a drum. In fact, the modes which most significantly determine the tone quality of a tympani drum are the (1,1), (2,1), (3,1), (4,1), and (5,1) modes.

The (0,2) Mode does not have any diameter nodes, but has two circular nodes - one at the outside edge and one at a distance of 0.436 a (a is the radius of the circular membrane) from the center of the membrane.

The frequency of the (0,2) mode is 2.295 times the frequency of the (0,1) mode. Like the (0,1) mode, the (0,2) mode is excited when the membrane is struck at the center. The sound radiation characteristics of the (0,2) mode are more complicated than the first three modes -- it appears to be a mix between a monopole and a dipole. Its decay time is longer than the (0,1) mode, but shorter than the (1,1) mode. As a result, it contributes to the "thump" sound when a drum is hit at the center, but does not contribute much to the musical pitch of a drum when hit off center.

The (1,2) Mode has one nodal diameter and two nodal circles.

The frequency of the (1,2) mode is 2.917 times the frequency of the (0,1) mode. As you might expect after looking at the first several modes of the circular membrane, the (1,2) mode does not radiate sound very effectively. It has somewhat of a quadrupole type behavior. Thus, the (1,2) mode takes a relatively long time to decay. However, this mode doesn't seem to play a dominant role in the musical tone quality of a drum.

The (0,3) Mode has three circular nodes, but no diameter nodes.

The frequency of the (0,3) mode is 3.598 times the frequency of the (0,1) mode. Like the (0,1) and (0,2) modes, the (0,3) mode is excited when the membrane is struck at the center. The sound radiation characteristics of the (0,3) mode rather complicated. This mode is excited when the membrane is struck at the center, and it dies away fairly quickly. As a result, it contributes to the "thump" sound when a drum is hit at the center, but does not contribute much to the musical pitch of a drum when hit off center.

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I think I need an advil.

Is that a wormhole? And the reason you have done is (an excellent job by the way) is because............

Made me kinda horny. But Angelo, I must be one of those average, or what ever you called 'em, people you spoke about in your boombastic thread. I'm a little lost, excited, but lost. marko

I don't find this thread the least bit funny. How dare you?

How do you generate 2,1 1,2 and 0,3 ?

BTW that's not crypt for "I want to meet Neil Peart" .

I hope you meant cryptic and not the other...

LOL I think crypt, meaning the specific language is allowable

Might even have the same meaning as the mausoleum one.

I think I need an advil.

I need a Dramamine.

Quit starin' at those humpy things.

Circular membrane = drumhead

This is about vibrational modes of drumheads, in other word the drumhead vibrates and radiates sound when the membrane is struck. Each musical instruments produces a spectrum of frequencies. Any instrument that uses a stretched membrane to produce sound is known as a membranophone. The head of a drum is its membrane. Because most drums have both a batter head and a resonant head they are more properly called bimembranophones.

Membranophones produce a more complex waveform than chordophones (stringed instruments) or aerophones (wind instruments). It is easy to imagine the air being moved back and forth two-hundred and twenty times per second when one plucks the A string on a guitar, but less simple to visualize the air being moved at the same frequency in a three-dimensional pattern off of a drumhead. Best most can do to visualize a vibrating membranophone is to think of bouncing on a trampoline.

When a guitar string is plucked, one observes that the string vibrates:

http://www.isvr.soton.ac.uk/SPCG/Tutorial/Tutorial/Tutorial_files/Web-standing-string.htm

More complicated vibrating systems such as membranes (drumheads) and plates (i.e. cymbals) also have vibrating modes and resonance frequencies:

http://www.isvr.soton.ac.uk/SPCG/Tutorial/Tutorial/Tutorial_files/Web-standing-membrane.htm

Research on the acoustics of percussion instruments has focussed on observing their modes of vibration and understanding how they radiate sound. Holographic interferometry, on account of its high resolution, is an especially useful method for modal analysis on a wide variety of percussion instruments. When the membrane (drumhead) is struck, many different modes or patterns of vibrations are excited.

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The five important modes

Each of these modes of vibration has its own pattern of radiation. The first (0.1) mode radiates pretty much in all directions (see Fig. 1 below), but the other four (1.1) 2.1) 3.1) and (4.1) modes radiate most strongly in 2, 4, 6, or 8 directions. They generate sound fields which are said to exhibit monopole, dipole, quadrupole, hexapole, and octupole character.

When the membrane is struck, many different modes or patterns of vibrations are excited, including the six important ones shown in Fig. 1. Each of these modes of vibration has its own pattern of radiation. The first one radiates pretty much in all directions, at least in its own plane, but the other four radiate most strongly in 2, 4, 6, or 8 directions, respectively. They generate sound fields which are said to exhibit monopole, dipole, quadrupole, hexapole, and octupole character.

Fig. 1 Five modes of vibration of a drumhead. Arrows indicate the directions of maximum sound radiation in the plane of the membrane:

In a two-headed drum, such as a snare drum, the directional radiation patterns from the two vibrating membranes interact, and the directionality of the sound becomes even more complex. In fact, the two membranes interact strongly as they vibrate, and so the modes or patterns of the drum are quite different form what they are in a drum with a single membrane.

Four modes of vibration of a snare drum, in which the heads vibrate in much the same way as they do in the first two modes in Fig. 1, are shown in Fig. 2. The directions in which maximum sound is radiated are indicated by arrows:

Fig. 2 Four modes of vibration of a two-headed snare drum. The heads vibrate in much the same patterns as the first two modes in Fig. 1. Directions of maximum radiation for each mode are shown by the arrows:

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Laws of modes

1. Each mode is excited more or less, according to how closely the pattern of the disturbing force matches the pattern of the mode itself.

2. Striking the membrane at any given point excites each natural mode in proportion to how much that mode involves motion of that particular point.

3. If the striking point is on a nodal point or line of some particular mode, then that mode is completely left out of the recipe (overtones/partials).

4. Striking an entire region at once instead of just a point produces the same recipe as you get by adding together the recipes for striking at each of the points contained in that region.

5. If a striking force has finite duration T in time (i.e. a single stroke with a drum stick), then only modes whose frequencies are less than about 2/T are efficiently excited.

6. Localized frictional damping will affect each mode in proportion to how much motion that mode causes at the point of application of the damping; in particular, it leaves undisturbed any mode that has a node at the point of application.

7. When a drumhead is struck sharply near the center, most of the energy initially appears in the (0,1) and (0,2) modes. By the end of the 1st second, their spectral peaks narrow substantially, and the sound spectrum includes many partials radiated by modes that received energy from the (0,1) and (0,2) modes to which they are coupled.

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Is this like a harmonic? my brain hurts..need..help:freak:

Yes, a harmonic by dampening, or in other word a flageolet on a drumhead.

The five important modes Fig. 2 Four modes of vibration of a two-headed snare drum. The heads vibrate in much the same patterns as the first two modes in Fig. 1. Directions of maximum radiation for each mode are shown by the arrows:

This assumes even tension? If so, should we not all be using 4 lug drums?

The tension is measured as well calculable. The vibrational modes are three dimensional and measured with interferometry. The drumhead (circular membrane) is a two dimensional object. The radiation is three dimensional.

Author References:

1)

I don't want to work, I just want to bang on the drums all day!

I don't find this thread the least bit funny. How dare you? How do you generate 2,1 1,2 and 0,3 ? BTW that's not crypt for "I want to meet Neil Peart" .

Take timpani lessons. I'm not sure about most lessons, but when I learned timpani, we had mode lessons, and how to strike the drum in different ways to excite specific modes to generate specific pitches and harmonies, because it is possible to have more than one mode going at one time.

This is why people like e-rings and moongels so much. Moongels can dampen all triangular shaped modes, e.g; 11, 21, 31, 41, 22, 51, 32, 61 and E-rings can dampen all circular shaped mode, e.g; 02, 12, 22, 03, 32. If you put a moongel on the nodal line of a circular shaped mode, but in the center of a triangular shaped mode, it will dampen the triangle mode, but not the circular mode.

The nice thing about mode when you don't use e-rings and moongels, is that you can use stick technique to change which mode is the one that can be heard.

For the modes in the first post, the frequency of the fundamental pitch (mode 01) is 0.766 divided by the diameter of the head, multiplied by the square root of the tension of the head divided by the density of the head. Tension measured in newtons per meter width, and density measured in kilograms per square meter. Multiply the fundamental pitch with the relative frequencies of the different modes, and you get the different pitches that are available from that particular drum.

Move the stick around and see how different strokes get different pitches and overtones. It is fun.

The radiation of a tympani

a = monopol, mode (01)

a' = monopol, mode (01)

b = dipol

b' = dipol, mode (01), free vibrating timapni head (no vessel)

b'' = dipol, mode (11)

c = quadrupole

c' = quadrupole, mode (11), free vibrating timapni head (no vessel)

c'' = quadrupole, mode (21)

.From Thomas D. Rossing, "The Acoustics of Tympani" 1978

Interesting. That last mode in the very first post in this thread looks EXACTLY like Kurt Vonnegut's asshole.

/w

Yow. Who'd you study with, Kraft? I never even imagined there was so much detail on drum striking and moon gelling. Thanks for the lead.

Yow. Who'd you study with, Kraft? I never even imagined there was so much detail on drum striking and moon gelling. Thanks for the lead.

Anyone who teaches timpani players how to get specific pitches should know the stick technique at the very least.

Never got that far into it. Tried American versions of French and German grips. Not that I get to play timpani but I settled on the Mitchell Peters/Eastman Universal grip. Big tone and intonation was as far as I got. This stuff with specific nodality, well I only get that picky on a snare drum and only by feel at that.